Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{-4t^3 - 4t^2 + 8t}{-4t^3 + 12t^2 + 40t}$
First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {-4t(t^2 + t - 2)} {-4t(t^2 - 3t - 10)} $ $ r = \dfrac{4t}{4t} \cdot \dfrac{t^2 + t - 2}{t^2 - 3t - 10} $ Simplify: $ r = \dfrac{t^2 + t - 2}{t^2 - 3t - 10}$ Since we are dividing by $t$ , we must remember that $t \neq 0$ Next factor the numerator and denominator. $ r = \dfrac{(t + 2)(t - 1)}{(t + 2)(t - 5)}$ Assuming $t \neq -2$ , we can cancel the $t + 2$ $ r = \dfrac{t - 1}{t - 5}$ Therefore: $ r = \dfrac{ t - 1 }{ t - 5 }$, $t \neq -2$, $t \neq 0$